Prove the following identities.
$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Verified step by step guidance

1. Start by recalling the definition of the tangent function in terms of sine and cosine: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). This is a fundamental trigonometric identity.
2. To prove the identity \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), we need to express \( \tan\theta \) in terms of \( \sin\theta \) and \( \cos\theta \).
3. The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the opposite side to the adjacent side. In terms of the unit circle, this translates to \( \tan\theta = \frac{y}{x} \), where \( y = \sin\theta \) and \( x = \cos\theta \).
4. Thus, substituting the unit circle definitions, we have \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), which matches the given identity.
5. Therefore, the identity \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) is proven using the definitions of sine, cosine, and tangent in the context of the unit circle.